通过多初值联合训练，提升神经微分方程对未知初始条件的泛化能力。
MPINeuralODE: Multiple-Initial-Condition Physics-Informed Neural ODEs for Globally Consistent Dynamical System Learning
- 采用多初值多射击策略，结合软物理约束优化动力系统建模。
- 在洛特卡-沃尔泰拉模型上，外推误差和长时程误差降低24%。
- 适合需要高稳定性与长期预测能力的动力系统学习任务。
神经常微分方程（Neural ODE）通常能拟合训练轨迹，但在未见初值和长时程预测上泛化能力差。本文提出MPINeuralODE，融合软物理约束与多初值（MIC）多射击课程训练，两者结构互补：物理项在MIC扩展的支持域上锚定向量场幅度。我们在三个维度评估：样本外误差、长时程稳定性与哈密顿漂移，共同检验学习到的动力系统是否恢复底层向量场。在洛特卡-沃尔泰拉模型上，MPINeuralODE在数据驱动方法中达到最低的样本外与长时程均方误差，相比基线Neural ODE降低24%，且哈密顿漂移表现接近PINN消融实验。
原文摘要 · Abstract (English)
Neural ordinary differential equations (Neural ODEs) often fit training trajectories while generalizing poorly to unseen initial conditions and long horizons. We propose MPINeuralODE, which combines a soft physics-informed residual with a Multiple-Initial-Condition (MIC) multiple-shooting curriculum whose ingredients are structurally complementary: the physics term anchors the vector-field magnitude on the support that MIC enlarges. We evaluate along three axes: out-of-sample error, long-horizon stability, and Hamiltonian drift, which together expose whether the learned dynamics recover the underlying vector field. On Lotka-Volterra, MPINeuralODE achieves the lowest out-of-sample and long-horizon MSE among data-driven methods, with a 24% reduction over the baseline Neural ODE, while essentially matching the PINN ablation on Hamiltonian drift.
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